217 



ration of the moon, i. e. about \, it has been shown in the " Connais- 

 sance du Temps," for 1800, that a variation in the length of the day of 

 about a ten-millionth of a second, or, roughly, of about a million mil- 

 lionth part of the entire length would suffice ; whereas, the value above 

 given only amounts to about a seventy millionth part of this. And in 

 like manner we shall find that the other terms mentioned, viz. those 

 depending upon the argument 6 X , will only produce quantities which 

 may be put down as quantities of the same order, and apparently not 

 more important than the above. The above effect is wholly due to the 

 moon ; that owing to the sun will be still smaller. Also, the earth 

 has been treated as a solid body, no account being taken of its being 

 partially covered with fluid. But Mr. Airy, I believe, has shown that 

 the effect of the friction of the tidal wave in altering the rotation is 

 quite insensible ; so that we may conclude that no effect sufficient to be 

 perceptible in 2000 years has been produced by the action of gravity ; 

 and hence direct perturbation of the rotatory motion is wholly inade- 

 quate to explain the phenomena mentioned. 



A possible explanation, however, may be given by supposing that 

 the slight alterations in figure which have been going on from geologi- 

 cal or other causes may have been of a kind that shall suffice to cause 

 such an alteration as would be required. If iv is the angular velocity, 

 it is well known that 



2w 2 



c being a constant, m and r the mass and distance from the centre of any 

 particle. Now, the mass remaining constant, it will be seen that a va- 

 riation in length of the radius of gyration amounting to about a three 

 thousandth part of an inch would be sufficient to produce the requisite 

 effect. It is no violent supposition to suppose that such a change may 

 have happened : perhaps the real difficulty would be, on the other 

 hand, to suppose that a body with a radius of 4000 miles had remained 

 so constant. But, on the other hand, the variation ought to be in in- 

 crease, not a diminution in the radius ; and there is nothing to show 

 that any changes that have taken place have tended in this direction 

 rather than the other, further than this : — that any departure from the 

 circular form in the sections parallel to the Equator tends to increase 

 the radius of gyration ; so that if the general effect of geological changes 

 has been to increase the inequalities of the surface, their effect would 

 also have been slightly to lengthen the day, and hence, also, to cause 

 an apparent acceleration of the moon. 

 (2.) Application to the Moon. 



The chief question of interest which here occurs is : " what are the 

 conditions of stable rotation?" or, in other words, what must be the re- 

 lation between n and n 1 in order that there may be no permanent varia- 

 tion in the rotation ? To answer this, it will be necessary to put down 



the general form which the equation for ^ will assume after the ap- 



